LMFDB - Embedded newform 850.4.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [850,4,Mod(1,850)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("850.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	850.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,10,5,20,0,10,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$50.1516235049$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 92x^{3} - 26x^{2} + 2010x + 702 $ x^5 - 92x^3 - 26x^2 + 2010*x + 702
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$7.79664$ of defining polynomial
Character	$\chi$	$=$	850.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +8.79664 q^{3} +4.00000 q^{4} +17.5933 q^{6} -24.8284 q^{7} +8.00000 q^{8} +50.3809 q^{9} +1.58122 q^{11} +35.1866 q^{12} +68.9939 q^{13} -49.6568 q^{14} +16.0000 q^{16} +17.0000 q^{17} +100.762 q^{18} +52.5163 q^{19} -218.407 q^{21} +3.16244 q^{22} +154.124 q^{23} +70.3731 q^{24} +137.988 q^{26} +205.674 q^{27} -99.3137 q^{28} -9.87891 q^{29} +133.948 q^{31} +32.0000 q^{32} +13.9094 q^{33} +34.0000 q^{34} +201.524 q^{36} -337.139 q^{37} +105.033 q^{38} +606.915 q^{39} -220.387 q^{41} -436.813 q^{42} -16.7940 q^{43} +6.32489 q^{44} +308.247 q^{46} +224.816 q^{47} +140.746 q^{48} +273.450 q^{49} +149.543 q^{51} +275.976 q^{52} -302.962 q^{53} +411.347 q^{54} -198.627 q^{56} +461.967 q^{57} -19.7578 q^{58} +469.158 q^{59} +776.705 q^{61} +267.895 q^{62} -1250.88 q^{63} +64.0000 q^{64} +27.8189 q^{66} +74.5265 q^{67} +68.0000 q^{68} +1355.77 q^{69} +418.017 q^{71} +403.047 q^{72} +1016.75 q^{73} -674.277 q^{74} +210.065 q^{76} -39.2592 q^{77} +1213.83 q^{78} -1360.36 q^{79} +448.952 q^{81} -440.774 q^{82} -625.149 q^{83} -873.627 q^{84} -33.5879 q^{86} -86.9012 q^{87} +12.6498 q^{88} -79.3572 q^{89} -1713.01 q^{91} +616.494 q^{92} +1178.29 q^{93} +449.632 q^{94} +281.493 q^{96} +537.519 q^{97} +546.901 q^{98} +79.6634 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{2} + 5 q^{3} + 20 q^{4} + 10 q^{6} - q^{7} + 40 q^{8} + 54 q^{9} + 38 q^{11} + 20 q^{12} + 93 q^{13} - 2 q^{14} + 80 q^{16} + 85 q^{17} + 108 q^{18} - 50 q^{19} + 55 q^{21} + 76 q^{22} + 98 q^{23}+ \cdots + 1006 q^{99}+O(q^{100}) $ 5 * q + 10 * q^2 + 5 * q^3 + 20 * q^4 + 10 * q^6 - q^7 + 40 * q^8 + 54 * q^9 + 38 * q^11 + 20 * q^12 + 93 * q^13 - 2 * q^14 + 80 * q^16 + 85 * q^17 + 108 * q^18 - 50 * q^19 + 55 * q^21 + 76 * q^22 + 98 * q^23 + 40 * q^24 + 186 * q^26 + 365 * q^27 - 4 * q^28 + 102 * q^29 + 149 * q^31 + 160 * q^32 + 28 * q^33 + 170 * q^34 + 216 * q^36 + 414 * q^37 - 100 * q^38 + 289 * q^39 + 68 * q^41 + 110 * q^42 - 180 * q^43 + 152 * q^44 + 196 * q^46 + 958 * q^47 + 80 * q^48 + 734 * q^49 + 85 * q^51 + 372 * q^52 - 171 * q^53 + 730 * q^54 - 8 * q^56 + 1564 * q^57 + 204 * q^58 + 404 * q^59 + 650 * q^61 + 298 * q^62 - 694 * q^63 + 320 * q^64 + 56 * q^66 + 1792 * q^67 + 340 * q^68 + 1568 * q^69 + 283 * q^71 + 432 * q^72 + 294 * q^73 + 828 * q^74 - 200 * q^76 + 2284 * q^77 + 578 * q^78 + 951 * q^79 - 1355 * q^81 + 136 * q^82 - 1468 * q^83 + 220 * q^84 - 360 * q^86 + 2544 * q^87 + 304 * q^88 + 1666 * q^89 - 915 * q^91 + 392 * q^92 - 147 * q^93 + 1916 * q^94 + 160 * q^96 + 2304 * q^97 + 1468 * q^98 + 1006 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	8.79664	1.69291	0.846457	−	0.532457i	$-0.178731\pi$
$3$	8.79664	1.69291	0.846457	+	0.532457i	$0.178731\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	17.5933	1.19707
$7$	−24.8284	−1.34061	−0.670304	−	0.742086i	$-0.733836\pi$
$7$	−24.8284	−1.34061	−0.670304	+	0.742086i	$0.733836\pi$
$8$	8.00000	0.353553
$9$	50.3809	1.86596
$10$	0	0
$11$	1.58122	0.0433415	0.0216707	−	0.999765i	$-0.493101\pi$
$11$	1.58122	0.0433415	0.0216707	+	0.999765i	$0.493101\pi$
$12$	35.1866	0.846457
$13$	68.9939	1.47196	0.735980	−	0.677004i	$-0.236722\pi$
$13$	68.9939	1.47196	0.735980	+	0.677004i	$0.236722\pi$
$14$	−49.6568	−0.947953
$15$	0	0
$16$	16.0000	0.250000
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	100.762	1.31943
$19$	52.5163	0.634109	0.317055	−	0.948407i	$-0.397306\pi$
$19$	52.5163	0.634109	0.317055	+	0.948407i	$0.397306\pi$
$20$	0	0
$21$	−218.407	−2.26954
$22$	3.16244	0.0306471
$23$	154.124	1.39726	0.698630	−	0.715483i	$-0.253793\pi$
$23$	154.124	1.39726	0.698630	+	0.715483i	$0.253793\pi$
$24$	70.3731	0.598536
$25$	0	0
$26$	137.988	1.04083
$27$	205.674	1.46600
$28$	−99.3137	−0.670304
$29$	−9.87891	−0.0632575	−0.0316287	−	0.999500i	$-0.510069\pi$
$29$	−9.87891	−0.0632575	−0.0316287	+	0.999500i	$0.510069\pi$
$30$	0	0
$31$	133.948	0.776055	0.388027	−	0.921648i	$-0.373157\pi$
$31$	133.948	0.776055	0.388027	+	0.921648i	$0.373157\pi$
$32$	32.0000	0.176777
$33$	13.9094	0.0733734
$34$	34.0000	0.171499
$35$	0	0
$36$	201.524	0.932980
$37$	−337.139	−1.49798	−0.748990	−	0.662581i	$-0.769461\pi$
$37$	−337.139	−1.49798	−0.748990	+	0.662581i	$0.769461\pi$
$38$	105.033	0.448383
$39$	606.915	2.49190
$40$	0	0
$41$	−220.387	−0.839479	−0.419739	−	0.907645i	$-0.637879\pi$
$41$	−220.387	−0.839479	−0.419739	+	0.907645i	$0.637879\pi$
$42$	−436.813	−1.60480
$43$	−16.7940	−0.0595594	−0.0297797	−	0.999556i	$-0.509481\pi$
$43$	−16.7940	−0.0595594	−0.0297797	+	0.999556i	$0.509481\pi$
$44$	6.32489	0.0216707
$45$	0	0
$46$	308.247	0.988013
$47$	224.816	0.697719	0.348859	−	0.937175i	$-0.386569\pi$
$47$	224.816	0.697719	0.348859	+	0.937175i	$0.386569\pi$
$48$	140.746	0.423229
$49$	273.450	0.797232
$50$	0	0
$51$	149.543	0.410592
$52$	275.976	0.735980
$53$	−302.962	−0.785190	−0.392595	−	0.919712i	$-0.628423\pi$
$53$	−302.962	−0.785190	−0.392595	+	0.919712i	$0.628423\pi$
$54$	411.347	1.03662
$55$	0	0
$56$	−198.627	−0.473977
$57$	461.967	1.07349
$58$	−19.7578	−0.0447298
$59$	469.158	1.03524	0.517620	−	0.855611i	$-0.326818\pi$
$59$	469.158	1.03524	0.517620	+	0.855611i	$0.326818\pi$
$60$	0	0
$61$	776.705	1.63028	0.815139	−	0.579266i	$-0.196661\pi$
$61$	776.705	1.63028	0.815139	+	0.579266i	$0.196661\pi$
$62$	267.895	0.548754
$63$	−1250.88	−2.50152
$64$	64.0000	0.125000
$65$	0	0
$66$	27.8189	0.0518829
$67$	74.5265	0.135894	0.0679468	−	0.997689i	$-0.478355\pi$
$67$	74.5265	0.135894	0.0679468	+	0.997689i	$0.478355\pi$
$68$	68.0000	0.121268
$69$	1355.77	2.36544
$70$	0	0
$71$	418.017	0.698726	0.349363	−	0.936987i	$-0.386398\pi$
$71$	418.017	0.698726	0.349363	+	0.936987i	$0.386398\pi$
$72$	403.047	0.659716
$73$	1016.75	1.63015	0.815077	−	0.579353i	$-0.196695\pi$
$73$	1016.75	1.63015	0.815077	+	0.579353i	$0.196695\pi$
$74$	−674.277	−1.05923
$75$	0	0
$76$	210.065	0.317055
$77$	−39.2592	−0.0581040
$78$	1213.83	1.76204
$79$	−1360.36	−1.93738	−0.968690	−	0.248275i	$-0.920136\pi$
$79$	−1360.36	−1.93738	−0.968690	+	0.248275i	$0.920136\pi$
$80$	0	0
$81$	448.952	0.615846
$82$	−440.774	−0.593601
$83$	−625.149	−0.826735	−0.413367	−	0.910564i	$-0.635647\pi$
$83$	−625.149	−0.826735	−0.413367	+	0.910564i	$0.635647\pi$
$84$	−873.627	−1.13477
$85$	0	0
$86$	−33.5879	−0.0421149
$87$	−86.9012	−0.107090
$88$	12.6498	0.0153235
$89$	−79.3572	−0.0945151	−0.0472576	−	0.998883i	$-0.515048\pi$
$89$	−79.3572	−0.0945151	−0.0472576	+	0.998883i	$0.515048\pi$
$90$	0	0
$91$	−1713.01	−1.97332
$92$	616.494	0.698630
$93$	1178.29	1.31379
$94$	449.632	0.493362
$95$	0	0
$96$	281.493	0.299268
$97$	537.519	0.562647	0.281323	−	0.959613i	$-0.409227\pi$
$97$	537.519	0.562647	0.281323	+	0.959613i	$0.409227\pi$
$98$	546.901	0.563728
$99$	79.6634	0.0808735
$100$	0	0
$101$	−545.388	−0.537308	−0.268654	−	0.963237i	$-0.586579\pi$
$101$	−545.388	−0.537308	−0.268654	+	0.963237i	$0.586579\pi$
$102$	299.086	0.290332
$103$	−1640.16	−1.56902	−0.784512	−	0.620114i	$-0.787086\pi$
$103$	−1640.16	−1.56902	−0.784512	+	0.620114i	$0.787086\pi$
$104$	551.951	0.520416
$105$	0	0
$106$	−605.924	−0.555213
$107$	2023.85	1.82853	0.914267	−	0.405113i	$-0.132768\pi$
$107$	2023.85	1.82853	0.914267	+	0.405113i	$0.132768\pi$
$108$	822.694	0.732998
$109$	−388.582	−0.341462	−0.170731	−	0.985318i	$-0.554613\pi$
$109$	−388.582	−0.341462	−0.170731	+	0.985318i	$0.554613\pi$
$110$	0	0
$111$	−2965.69	−2.53595
$112$	−397.255	−0.335152
$113$	−1844.72	−1.53572	−0.767862	−	0.640615i	$-0.778679\pi$
$113$	−1844.72	−1.53572	−0.767862	+	0.640615i	$0.778679\pi$
$114$	923.935	0.759074
$115$	0	0
$116$	−39.5156	−0.0316287
$117$	3475.98	2.74662
$118$	938.315	0.732025
$119$	−422.083	−0.325145
$120$	0	0
$121$	−1328.50	−0.998122
$122$	1553.41	1.15278
$123$	−1938.66	−1.42117
$124$	535.790	0.388027
$125$	0	0
$126$	−2501.76	−1.76884
$127$	−1094.46	−0.764703	−0.382352	−	0.924017i	$-0.624886\pi$
$127$	−1094.46	−0.764703	−0.382352	+	0.924017i	$0.624886\pi$
$128$	128.000	0.0883883
$129$	−147.730	−0.100829
$130$	0	0
$131$	−2045.28	−1.36410	−0.682051	−	0.731305i	$-0.738912\pi$
$131$	−2045.28	−1.36410	−0.682051	+	0.731305i	$0.738912\pi$
$132$	55.6378	0.0366867
$133$	−1303.90	−0.850092
$134$	149.053	0.0960912
$135$	0	0
$136$	136.000	0.0857493
$137$	−901.223	−0.562020	−0.281010	−	0.959705i	$-0.590669\pi$
$137$	−901.223	−0.562020	−0.281010	+	0.959705i	$0.590669\pi$
$138$	2711.54	1.67262
$139$	2117.29	1.29199	0.645993	−	0.763343i	$-0.276443\pi$
$139$	2117.29	1.29199	0.645993	+	0.763343i	$0.276443\pi$
$140$	0	0
$141$	1977.63	1.18118
$142$	836.035	0.494074
$143$	109.095	0.0637969
$144$	806.095	0.466490
$145$	0	0
$146$	2033.49	1.15269
$147$	2405.45	1.34964
$148$	−1348.55	−0.748990
$149$	−1667.38	−0.916759	−0.458379	−	0.888757i	$-0.651570\pi$
$149$	−1667.38	−0.916759	−0.458379	+	0.888757i	$0.651570\pi$
$150$	0	0
$151$	−3000.39	−1.61701	−0.808505	−	0.588489i	$-0.799723\pi$
$151$	−3000.39	−1.61701	−0.808505	+	0.588489i	$0.799723\pi$
$152$	420.131	0.224191
$153$	856.476	0.452562
$154$	−78.5185	−0.0410857
$155$	0	0
$156$	2427.66	1.24595
$157$	−2746.77	−1.39628	−0.698140	−	0.715961i	$-0.745989\pi$
$157$	−2746.77	−1.39628	−0.698140	+	0.715961i	$0.745989\pi$
$158$	−2720.73	−1.36993
$159$	−2665.05	−1.32926
$160$	0	0
$161$	−3826.65	−1.87318
$162$	897.903	0.435469
$163$	716.666	0.344378	0.172189	−	0.985064i	$-0.444916\pi$
$163$	716.666	0.344378	0.172189	+	0.985064i	$0.444916\pi$
$164$	−881.547	−0.419739
$165$	0	0
$166$	−1250.30	−0.584590
$167$	2612.11	1.21037	0.605183	−	0.796086i	$-0.293100\pi$
$167$	2612.11	1.21037	0.605183	+	0.796086i	$0.293100\pi$
$168$	−1747.25	−0.802402
$169$	2563.16	1.16666
$170$	0	0
$171$	2645.82	1.18322
$172$	−67.1758	−0.0297797
$173$	210.723	0.0926066	0.0463033	−	0.998927i	$-0.485256\pi$
$173$	210.723	0.0926066	0.0463033	+	0.998927i	$0.485256\pi$
$174$	−173.802	−0.0757237
$175$	0	0
$176$	25.2996	0.0108354
$177$	4127.01	1.75257
$178$	−158.714	−0.0668323
$179$	−2078.07	−0.867721	−0.433861	−	0.900980i	$-0.642849\pi$
$179$	−2078.07	−0.867721	−0.433861	+	0.900980i	$0.642849\pi$
$180$	0	0
$181$	4233.95	1.73871	0.869357	−	0.494185i	$-0.164533\pi$
$181$	4233.95	1.73871	0.869357	+	0.494185i	$0.164533\pi$
$182$	−3426.02	−1.39535
$183$	6832.40	2.75992
$184$	1232.99	0.494006
$185$	0	0
$186$	2356.58	0.928993
$187$	26.8808	0.0105119
$188$	899.264	0.348859
$189$	−5106.55	−1.96533
$190$	0	0
$191$	−2824.24	−1.06992	−0.534960	−	0.844877i	$-0.679673\pi$
$191$	−2824.24	−1.06992	−0.534960	+	0.844877i	$0.679673\pi$
$192$	562.985	0.211614
$193$	3005.32	1.12087	0.560435	−	0.828199i	$-0.310634\pi$
$193$	3005.32	1.12087	0.560435	+	0.828199i	$0.310634\pi$
$194$	1075.04	0.397851
$195$	0	0
$196$	1093.80	0.398616
$197$	506.506	0.183183	0.0915915	−	0.995797i	$-0.470805\pi$
$197$	506.506	0.183183	0.0915915	+	0.995797i	$0.470805\pi$
$198$	159.327	0.0571862
$199$	−3408.26	−1.21410	−0.607048	−	0.794665i	$-0.707646\pi$
$199$	−3408.26	−1.21410	−0.607048	+	0.794665i	$0.707646\pi$
$200$	0	0
$201$	655.583	0.230056
$202$	−1090.78	−0.379934
$203$	245.278	0.0848035
$204$	598.172	0.205296
$205$	0	0
$206$	−3280.31	−1.10947
$207$	7764.89	2.60723
$208$	1103.90	0.367990
$209$	83.0400	0.0274832
$210$	0	0
$211$	−3024.27	−0.986728	−0.493364	−	0.869823i	$-0.664233\pi$
$211$	−3024.27	−0.986728	−0.493364	+	0.869823i	$0.664233\pi$
$212$	−1211.85	−0.392595
$213$	3677.15	1.18288
$214$	4047.70	1.29297
$215$	0	0
$216$	1645.39	0.518308
$217$	−3325.71	−1.04039
$218$	−777.163	−0.241450
$219$	8943.96	2.75971
$220$	0	0
$221$	1172.90	0.357003
$222$	−5931.38	−1.79319
$223$	431.326	0.129523	0.0647617	−	0.997901i	$-0.479371\pi$
$223$	431.326	0.129523	0.0647617	+	0.997901i	$0.479371\pi$
$224$	−794.509	−0.236988
$225$	0	0
$226$	−3689.44	−1.08592
$227$	−4991.95	−1.45959	−0.729796	−	0.683665i	$-0.760385\pi$
$227$	−4991.95	−1.45959	−0.729796	+	0.683665i	$0.760385\pi$
$228$	1847.87	0.536746
$229$	−3411.44	−0.984429	−0.492215	−	0.870474i	$-0.663812\pi$
$229$	−3411.44	−0.984429	−0.492215	+	0.870474i	$0.663812\pi$
$230$	0	0
$231$	−345.350	−0.0983651
$232$	−79.0313	−0.0223649
$233$	27.8076	0.00781861	0.00390931	−	0.999992i	$-0.498756\pi$
$233$	27.8076	0.00781861	0.00390931	+	0.999992i	$0.498756\pi$
$234$	6951.95	1.94215
$235$	0	0
$236$	1876.63	0.517620
$237$	−11966.6	−3.27982
$238$	−844.166	−0.229912
$239$	−5944.89	−1.60897	−0.804483	−	0.593976i	$-0.797557\pi$
$239$	−5944.89	−1.60897	−0.804483	+	0.593976i	$0.797557\pi$
$240$	0	0
$241$	7105.16	1.89910	0.949551	−	0.313611i	$-0.101539\pi$
$241$	7105.16	1.89910	0.949551	+	0.313611i	$0.101539\pi$
$242$	−2657.00	−0.705778
$243$	−1603.92	−0.423421
$244$	3106.82	0.815139
$245$	0	0
$246$	−3877.33	−1.00492
$247$	3623.31	0.933383
$248$	1071.58	0.274377
$249$	−5499.21	−1.39959
$250$	0	0
$251$	−1665.30	−0.418776	−0.209388	−	0.977833i	$-0.567147\pi$
$251$	−1665.30	−0.418776	−0.209388	+	0.977833i	$0.567147\pi$
$252$	−5003.51	−1.25076
$253$	243.704	0.0605594
$254$	−2188.91	−0.540727
$255$	0	0
$256$	256.000	0.0625000
$257$	2331.39	0.565868	0.282934	−	0.959139i	$-0.408692\pi$
$257$	2331.39	0.565868	0.282934	+	0.959139i	$0.408692\pi$
$258$	−295.461	−0.0712969
$259$	8370.62	2.00820
$260$	0	0
$261$	−497.708	−0.118036
$262$	−4090.57	−0.964566
$263$	5967.72	1.39918	0.699592	−	0.714543i	$-0.253365\pi$
$263$	5967.72	1.39918	0.699592	+	0.714543i	$0.253365\pi$
$264$	111.276	0.0259414
$265$	0	0
$266$	−2607.80	−0.601106
$267$	−698.077	−0.160006
$268$	298.106	0.0679468
$269$	4857.57	1.10101	0.550504	−	0.834833i	$-0.314436\pi$
$269$	4857.57	1.10101	0.550504	+	0.834833i	$0.314436\pi$
$270$	0	0
$271$	−1029.78	−0.230828	−0.115414	−	0.993317i	$-0.536820\pi$
$271$	−1029.78	−0.230828	−0.115414	+	0.993317i	$0.536820\pi$
$272$	272.000	0.0606339
$273$	−15068.7	−3.34066
$274$	−1802.45	−0.397408
$275$	0	0
$276$	5423.08	1.18272
$277$	−1411.71	−0.306214	−0.153107	−	0.988210i	$-0.548928\pi$
$277$	−1411.71	−0.306214	−0.153107	+	0.988210i	$0.548928\pi$
$278$	4234.58	0.913572
$279$	6748.40	1.44809
$280$	0	0
$281$	−4204.24	−0.892542	−0.446271	−	0.894898i	$-0.647248\pi$
$281$	−4204.24	−0.892542	−0.446271	+	0.894898i	$0.647248\pi$
$282$	3955.25	0.835219
$283$	4178.89	0.877771	0.438886	−	0.898543i	$-0.355373\pi$
$283$	4178.89	0.877771	0.438886	+	0.898543i	$0.355373\pi$
$284$	1672.07	0.349363
$285$	0	0
$286$	218.189	0.0451112
$287$	5471.86	1.12541
$288$	1612.19	0.329858
$289$	289.000	0.0588235
$290$	0	0
$291$	4728.36	0.952513
$292$	4066.99	0.815077
$293$	1114.73	0.222263	0.111132	−	0.993806i	$-0.464552\pi$
$293$	1114.73	0.222263	0.111132	+	0.993806i	$0.464552\pi$
$294$	4810.89	0.954343
$295$	0	0
$296$	−2697.11	−0.529616
$297$	325.215	0.0635384
$298$	−3334.76	−0.648246
$299$	10633.6	2.05671
$300$	0	0
$301$	416.968	0.0798459
$302$	−6000.79	−1.14340
$303$	−4797.58	−0.909617
$304$	840.262	0.158527
$305$	0	0
$306$	1712.95	0.320009
$307$	7999.20	1.48710	0.743548	−	0.668683i	$-0.233142\pi$
$307$	7999.20	1.48710	0.743548	+	0.668683i	$0.233142\pi$
$308$	−157.037	−0.0290520
$309$	−14427.9	−2.65622
$310$	0	0
$311$	−10096.7	−1.84093	−0.920466	−	0.390822i	$-0.872191\pi$
$311$	−10096.7	−1.84093	−0.920466	+	0.390822i	$0.872191\pi$
$312$	4855.32	0.881020
$313$	5552.78	1.00275	0.501376	−	0.865229i	$-0.332827\pi$
$313$	5552.78	1.00275	0.501376	+	0.865229i	$0.332827\pi$
$314$	−5493.54	−0.987319
$315$	0	0
$316$	−5441.46	−0.968690
$317$	−4534.80	−0.803470	−0.401735	−	0.915756i	$-0.631593\pi$
$317$	−4534.80	−0.803470	−0.401735	+	0.915756i	$0.631593\pi$
$318$	−5330.10	−0.939928
$319$	−15.6207	−0.00274167
$320$	0	0
$321$	17803.1	3.09555
$322$	−7653.29	−1.32454
$323$	892.778	0.153794
$324$	1795.81	0.307923
$325$	0	0
$326$	1433.33	0.243512
$327$	−3418.21	−0.578066
$328$	−1763.09	−0.296801
$329$	−5581.83	−0.935368
$330$	0	0
$331$	3365.40	0.558850	0.279425	−	0.960168i	$-0.409856\pi$
$331$	3365.40	0.558850	0.279425	+	0.960168i	$0.409856\pi$
$332$	−2500.59	−0.413367
$333$	−16985.4	−2.79517
$334$	5224.22	0.855858
$335$	0	0
$336$	−3494.51	−0.567384
$337$	724.433	0.117099	0.0585495	−	0.998285i	$-0.481352\pi$
$337$	724.433	0.117099	0.0585495	+	0.998285i	$0.481352\pi$
$338$	5126.32	0.824956
$339$	−16227.4	−2.59985
$340$	0	0
$341$	211.801	0.0336354
$342$	5291.64	0.836665
$343$	1726.81	0.271833
$344$	−134.352	−0.0210574
$345$	0	0
$346$	421.445	0.0654827
$347$	5740.27	0.888051	0.444025	−	0.896014i	$-0.353550\pi$
$347$	5740.27	0.888051	0.444025	+	0.896014i	$0.353550\pi$
$348$	−347.605	−0.0535448
$349$	−9374.12	−1.43778	−0.718890	−	0.695124i	$-0.755350\pi$
$349$	−9374.12	−1.43778	−0.718890	+	0.695124i	$0.755350\pi$
$350$	0	0
$351$	14190.2	2.15789
$352$	50.5991	0.00766177
$353$	3770.40	0.568493	0.284247	−	0.958751i	$-0.408257\pi$
$353$	3770.40	0.568493	0.284247	+	0.958751i	$0.408257\pi$
$354$	8254.02	1.23926
$355$	0	0
$356$	−317.429	−0.0472576
$357$	−3712.91	−0.550443
$358$	−4156.14	−0.613572
$359$	−3950.97	−0.580848	−0.290424	−	0.956898i	$-0.593796\pi$
$359$	−3950.97	−0.580848	−0.290424	+	0.956898i	$0.593796\pi$
$360$	0	0
$361$	−4101.03	−0.597905
$362$	8467.90	1.22946
$363$	−11686.3	−1.68973
$364$	−6852.04	−0.986661
$365$	0	0
$366$	13664.8	1.95156
$367$	−13582.6	−1.93190	−0.965949	−	0.258732i	$-0.916695\pi$
$367$	−13582.6	−1.93190	−0.965949	+	0.258732i	$0.916695\pi$
$368$	2465.98	0.349315
$369$	−11103.3	−1.56643
$370$	0	0
$371$	7522.07	1.05263
$372$	4713.16	0.656897
$373$	−2845.26	−0.394965	−0.197483	−	0.980306i	$-0.563277\pi$
$373$	−2845.26	−0.394965	−0.197483	+	0.980306i	$0.563277\pi$
$374$	53.7615	0.00743300
$375$	0	0
$376$	1798.53	0.246681
$377$	−681.584	−0.0931124
$378$	−10213.1	−1.38970
$379$	571.063	0.0773972	0.0386986	−	0.999251i	$-0.487679\pi$
$379$	571.063	0.0773972	0.0386986	+	0.999251i	$0.487679\pi$
$380$	0	0
$381$	−9627.54	−1.29458
$382$	−5648.48	−0.756548
$383$	9009.11	1.20194	0.600972	−	0.799270i	$-0.294780\pi$
$383$	9009.11	1.20194	0.600972	+	0.799270i	$0.294780\pi$
$384$	1125.97	0.149634
$385$	0	0
$386$	6010.64	0.792574
$387$	−846.095	−0.111135
$388$	2150.07	0.281323
$389$	1139.57	0.148530	0.0742651	−	0.997239i	$-0.476339\pi$
$389$	1139.57	0.148530	0.0742651	+	0.997239i	$0.476339\pi$
$390$	0	0
$391$	2620.10	0.338886
$392$	2187.60	0.281864
$393$	−17991.6	−2.30931
$394$	1013.01	0.129530
$395$	0	0
$396$	318.654	0.0404367
$397$	−1822.08	−0.230347	−0.115173	−	0.993345i	$-0.536742\pi$
$397$	−1822.08	−0.230347	−0.115173	+	0.993345i	$0.536742\pi$
$398$	−6816.52	−0.858495
$399$	−11469.9	−1.43913
$400$	0	0
$401$	13638.6	1.69845	0.849225	−	0.528031i	$-0.177069\pi$
$401$	13638.6	1.69845	0.849225	+	0.528031i	$0.177069\pi$
$402$	1311.17	0.162674
$403$	9241.57	1.14232
$404$	−2181.55	−0.268654
$405$	0	0
$406$	490.555	0.0599651
$407$	−533.091	−0.0649247
$408$	1196.34	0.145166
$409$	−3398.80	−0.410905	−0.205452	−	0.978667i	$-0.565867\pi$
$409$	−3398.80	−0.410905	−0.205452	+	0.978667i	$0.565867\pi$
$410$	0	0
$411$	−7927.74	−0.951451
$412$	−6560.63	−0.784512
$413$	−11648.4	−1.38785
$414$	15529.8	1.84359
$415$	0	0
$416$	2207.81	0.260208
$417$	18625.0	2.18722
$418$	166.080	0.0194336
$419$	−12623.8	−1.47186	−0.735932	−	0.677056i	$-0.763256\pi$
$419$	−12623.8	−1.47186	−0.735932	+	0.677056i	$0.763256\pi$
$420$	0	0
$421$	−677.408	−0.0784201	−0.0392100	−	0.999231i	$-0.512484\pi$
$421$	−677.408	−0.0784201	−0.0392100	+	0.999231i	$0.512484\pi$
$422$	−6048.55	−0.697722
$423$	11326.4	1.30192
$424$	−2423.70	−0.277606
$425$	0	0
$426$	7354.30	0.836425
$427$	−19284.4	−2.18556
$428$	8095.40	0.914267
$429$	959.667	0.108003
$430$	0	0
$431$	−3486.97	−0.389702	−0.194851	−	0.980833i	$-0.562422\pi$
$431$	−3486.97	−0.389702	−0.194851	+	0.980833i	$0.562422\pi$
$432$	3290.78	0.366499
$433$	3295.21	0.365722	0.182861	−	0.983139i	$-0.441464\pi$
$433$	3295.21	0.365722	0.182861	+	0.983139i	$0.441464\pi$
$434$	−6651.42	−0.735664
$435$	0	0
$436$	−1554.33	−0.170731
$437$	8094.01	0.886016
$438$	17887.9	1.95141
$439$	15136.6	1.64563	0.822816	−	0.568308i	$-0.192402\pi$
$439$	15136.6	1.64563	0.822816	+	0.568308i	$0.192402\pi$
$440$	0	0
$441$	13776.7	1.48760
$442$	2345.79	0.252439
$443$	57.2712	0.00614229	0.00307115	−	0.999995i	$-0.499022\pi$
$443$	57.2712	0.00614229	0.00307115	+	0.999995i	$0.499022\pi$
$444$	−11862.8	−1.26798
$445$	0	0
$446$	862.651	0.0915868
$447$	−14667.3	−1.55199
$448$	−1589.02	−0.167576
$449$	4022.35	0.422776	0.211388	−	0.977402i	$-0.432202\pi$
$449$	4022.35	0.422776	0.211388	+	0.977402i	$0.432202\pi$
$450$	0	0
$451$	−348.480	−0.0363843
$452$	−7378.89	−0.767862
$453$	−26393.4	−2.73746
$454$	−9983.90	−1.03209
$455$	0	0
$456$	3695.74	0.379537
$457$	−3310.52	−0.338861	−0.169431	−	0.985542i	$-0.554193\pi$
$457$	−3310.52	−0.338861	−0.169431	+	0.985542i	$0.554193\pi$
$458$	−6822.88	−0.696097
$459$	3496.45	0.355556
$460$	0	0
$461$	−5277.19	−0.533152	−0.266576	−	0.963814i	$-0.585892\pi$
$461$	−5277.19	−0.533152	−0.266576	+	0.963814i	$0.585892\pi$
$462$	−690.699	−0.0695546
$463$	−12572.6	−1.26198	−0.630992	−	0.775789i	$-0.717352\pi$
$463$	−12572.6	−1.26198	−0.630992	+	0.775789i	$0.717352\pi$
$464$	−158.063	−0.0158144
$465$	0	0
$466$	55.6152	0.00552859
$467$	3141.77	0.311314	0.155657	−	0.987811i	$-0.450251\pi$
$467$	3141.77	0.311314	0.155657	+	0.987811i	$0.450251\pi$
$468$	13903.9	1.37331
$469$	−1850.38	−0.182180
$470$	0	0
$471$	−24162.3	−2.36378
$472$	3753.26	0.366012
$473$	−26.5550	−0.00258139
$474$	−23933.3	−2.31918
$475$	0	0
$476$	−1688.33	−0.162573
$477$	−15263.5	−1.46513
$478$	−11889.8	−1.13771
$479$	−10023.2	−0.956101	−0.478051	−	0.878332i	$-0.658656\pi$
$479$	−10023.2	−0.956101	−0.478051	+	0.878332i	$0.658656\pi$
$480$	0	0
$481$	−23260.5	−2.20496
$482$	14210.3	1.34287
$483$	−33661.6	−3.17113
$484$	−5314.00	−0.499061
$485$	0	0
$486$	−3207.84	−0.299404
$487$	4236.72	0.394218	0.197109	−	0.980382i	$-0.436845\pi$
$487$	4236.72	0.394218	0.197109	+	0.980382i	$0.436845\pi$
$488$	6213.64	0.576390
$489$	6304.26	0.583003
$490$	0	0
$491$	−8275.32	−0.760611	−0.380305	−	0.924861i	$-0.624181\pi$
$491$	−8275.32	−0.760611	−0.380305	+	0.924861i	$0.624181\pi$
$492$	−7754.65	−0.710583
$493$	−167.941	−0.0153422
$494$	7246.62	0.660001
$495$	0	0
$496$	2143.16	0.194014
$497$	−10378.7	−0.936718
$498$	−10998.4	−0.989660
$499$	−10653.5	−0.955742	−0.477871	−	0.878430i	$-0.658591\pi$
$499$	−10653.5	−0.955742	−0.477871	+	0.878430i	$0.658591\pi$
$500$	0	0
$501$	22977.8	2.04905
$502$	−3330.60	−0.296119
$503$	21166.5	1.87627	0.938137	−	0.346264i	$-0.112550\pi$
$503$	21166.5	1.87627	0.938137	+	0.346264i	$0.112550\pi$
$504$	−10007.0	−0.884421
$505$	0	0
$506$	487.407	0.0428219
$507$	22547.2	1.97506
$508$	−4377.83	−0.382352
$509$	14591.0	1.27060	0.635298	−	0.772267i	$-0.280877\pi$
$509$	14591.0	1.27060	0.635298	+	0.772267i	$0.280877\pi$
$510$	0	0
$511$	−25244.2	−2.18540
$512$	512.000	0.0441942
$513$	10801.2	0.929601
$514$	4662.78	0.400129
$515$	0	0
$516$	−590.922	−0.0504145
$517$	355.484	0.0302402
$518$	16741.2	1.42002
$519$	1853.65	0.156775
$520$	0	0
$521$	9004.04	0.757148	0.378574	−	0.925571i	$-0.376414\pi$
$521$	9004.04	0.757148	0.378574	+	0.925571i	$0.376414\pi$
$522$	−995.417	−0.0834640
$523$	3937.43	0.329200	0.164600	−	0.986360i	$-0.447367\pi$
$523$	3937.43	0.329200	0.164600	+	0.986360i	$0.447367\pi$
$524$	−8181.14	−0.682051
$525$	0	0
$526$	11935.4	0.989372
$527$	2277.11	0.188221
$528$	222.551	0.0183434
$529$	11587.1	0.952338
$530$	0	0
$531$	23636.6	1.93171
$532$	−5215.59	−0.425046
$533$	−15205.3	−1.23568
$534$	−1396.15	−0.113141
$535$	0	0
$536$	596.212	0.0480456
$537$	−18280.0	−1.46898
$538$	9715.14	0.778530
$539$	432.386	0.0345532
$540$	0	0
$541$	−12395.2	−0.985045	−0.492522	−	0.870300i	$-0.663925\pi$
$541$	−12395.2	−0.985045	−0.492522	+	0.870300i	$0.663925\pi$
$542$	−2059.55	−0.163220
$543$	37244.6	2.94349
$544$	544.000	0.0428746
$545$	0	0
$546$	−30137.5	−2.36221
$547$	7862.53	0.614584	0.307292	−	0.951615i	$-0.400577\pi$
$547$	7862.53	0.614584	0.307292	+	0.951615i	$0.400577\pi$
$548$	−3604.89	−0.281010
$549$	39131.1	3.04203
$550$	0	0
$551$	−518.804	−0.0401122
$552$	10846.2	0.836310
$553$	33775.7	2.59727
$554$	−2823.41	−0.216526
$555$	0	0
$556$	8469.16	0.645993
$557$	18015.6	1.37046	0.685230	−	0.728327i	$-0.259702\pi$
$557$	18015.6	1.37046	0.685230	+	0.728327i	$0.259702\pi$
$558$	13496.8	1.02395
$559$	−1158.68	−0.0876690
$560$	0	0
$561$	236.461	0.0177957
$562$	−8408.49	−0.631122
$563$	1038.98	0.0777759	0.0388879	−	0.999244i	$-0.487618\pi$
$563$	1038.98	0.0777759	0.0388879	+	0.999244i	$0.487618\pi$
$564$	7910.50	0.590589
$565$	0	0
$566$	8357.78	0.620678
$567$	−11146.8	−0.825608
$568$	3344.14	0.247037
$569$	17635.2	1.29931	0.649653	−	0.760231i	$-0.274914\pi$
$569$	17635.2	1.29931	0.649653	+	0.760231i	$0.274914\pi$
$570$	0	0
$571$	−17239.2	−1.26347	−0.631733	−	0.775186i	$-0.717656\pi$
$571$	−17239.2	−1.26347	−0.631733	+	0.775186i	$0.717656\pi$
$572$	436.379	0.0318984
$573$	−24843.8	−1.81128
$574$	10943.7	0.795787
$575$	0	0
$576$	3224.38	0.233245
$577$	4212.23	0.303912	0.151956	−	0.988387i	$-0.451443\pi$
$577$	4212.23	0.303912	0.151956	+	0.988387i	$0.451443\pi$
$578$	578.000	0.0415945
$579$	26436.7	1.89754
$580$	0	0
$581$	15521.4	1.10833
$582$	9456.72	0.673529
$583$	−479.050	−0.0340313
$584$	8133.98	0.576346
$585$	0	0
$586$	2229.46	0.157164
$587$	15777.1	1.10936	0.554678	−	0.832065i	$-0.312841\pi$
$587$	15777.1	1.10936	0.554678	+	0.832065i	$0.312841\pi$
$588$	9621.78	0.674822
$589$	7034.44	0.492104
$590$	0	0
$591$	4455.55	0.310113
$592$	−5394.22	−0.374495
$593$	20030.2	1.38709	0.693543	−	0.720416i	$-0.256049\pi$
$593$	20030.2	1.38709	0.693543	+	0.720416i	$0.256049\pi$
$594$	650.431	0.0449285
$595$	0	0
$596$	−6669.52	−0.458379
$597$	−29981.2	−2.05536
$598$	21267.2	1.45431
$599$	−13934.8	−0.950517	−0.475259	−	0.879846i	$-0.657646\pi$
$599$	−13934.8	−0.950517	−0.475259	+	0.879846i	$0.657646\pi$
$600$	0	0
$601$	−3831.29	−0.260036	−0.130018	−	0.991512i	$-0.541503\pi$
$601$	−3831.29	−0.260036	−0.130018	+	0.991512i	$0.541503\pi$
$602$	833.935	0.0564596
$603$	3754.71	0.253572
$604$	−12001.6	−0.808505
$605$	0	0
$606$	−9595.16	−0.643196
$607$	4838.13	0.323515	0.161758	−	0.986831i	$-0.448284\pi$
$607$	4838.13	0.323515	0.161758	+	0.986831i	$0.448284\pi$
$608$	1680.52	0.112096
$609$	2157.62	0.143565
$610$	0	0
$611$	15510.9	1.02701
$612$	3425.90	0.226281
$613$	7554.80	0.497774	0.248887	−	0.968532i	$-0.419935\pi$
$613$	7554.80	0.497774	0.248887	+	0.968532i	$0.419935\pi$
$614$	15998.4	1.05154
$615$	0	0
$616$	−314.074	−0.0205429
$617$	−25836.5	−1.68580	−0.842899	−	0.538072i	$-0.819153\pi$
$617$	−25836.5	−1.68580	−0.842899	+	0.538072i	$0.819153\pi$
$618$	−28855.7	−1.87823
$619$	9216.86	0.598476	0.299238	−	0.954178i	$-0.403267\pi$
$619$	9216.86	0.598476	0.299238	+	0.954178i	$0.403267\pi$
$620$	0	0
$621$	31699.1	2.04838
$622$	−20193.3	−1.30174
$623$	1970.31	0.126708
$624$	9710.64	0.622975
$625$	0	0
$626$	11105.6	0.709053
$627$	730.473	0.0465268
$628$	−10987.1	−0.698140
$629$	−5731.36	−0.363313
$630$	0	0
$631$	26639.5	1.68067	0.840335	−	0.542068i	$-0.182358\pi$
$631$	26639.5	1.68067	0.840335	+	0.542068i	$0.182358\pi$
$632$	−10882.9	−0.684967
$633$	−26603.4	−1.67045
$634$	−9069.60	−0.568139
$635$	0	0
$636$	−10660.2	−0.664629
$637$	18866.4	1.17349
$638$	−31.2415	−0.00193866
$639$	21060.1	1.30379
$640$	0	0
$641$	−4546.25	−0.280135	−0.140067	−	0.990142i	$-0.544732\pi$
$641$	−4546.25	−0.280135	−0.140067	+	0.990142i	$0.544732\pi$
$642$	35606.2	2.18888
$643$	−27213.1	−1.66902	−0.834510	−	0.550993i	$-0.814249\pi$
$643$	−27213.1	−1.66902	−0.834510	+	0.550993i	$0.814249\pi$
$644$	−15306.6	−0.936590
$645$	0	0
$646$	1785.56	0.108749
$647$	−24147.2	−1.46727	−0.733634	−	0.679544i	$-0.762177\pi$
$647$	−24147.2	−1.46727	−0.733634	+	0.679544i	$0.762177\pi$
$648$	3591.61	0.217734
$649$	741.842	0.0448688
$650$	0	0
$651$	−29255.1	−1.76128
$652$	2866.66	0.172189
$653$	18096.1	1.08446	0.542232	−	0.840229i	$-0.317579\pi$
$653$	18096.1	1.08446	0.542232	+	0.840229i	$0.317579\pi$
$654$	−6836.43	−0.408754
$655$	0	0
$656$	−3526.19	−0.209870
$657$	51224.6	3.04180
$658$	−11163.7	−0.661405
$659$	−6305.57	−0.372732	−0.186366	−	0.982480i	$-0.559671\pi$
$659$	−6305.57	−0.372732	−0.186366	+	0.982480i	$0.559671\pi$
$660$	0	0
$661$	8546.53	0.502907	0.251454	−	0.967869i	$-0.419091\pi$
$661$	8546.53	0.502907	0.251454	+	0.967869i	$0.419091\pi$
$662$	6730.81	0.395166
$663$	10317.6	0.604375
$664$	−5001.19	−0.292295
$665$	0	0
$666$	−33970.7	−1.97648
$667$	−1522.57	−0.0883872
$668$	10448.4	0.605183
$669$	3794.22	0.219272
$670$	0	0
$671$	1228.14	0.0706586
$672$	−6989.01	−0.401201
$673$	23683.1	1.35649	0.678244	−	0.734836i	$-0.262741\pi$
$673$	23683.1	1.35649	0.678244	+	0.734836i	$0.262741\pi$
$674$	1448.87	0.0828015
$675$	0	0
$676$	10252.6	0.583332
$677$	−11419.3	−0.648271	−0.324136	−	0.946011i	$-0.605073\pi$
$677$	−11419.3	−0.648271	−0.324136	+	0.946011i	$0.605073\pi$
$678$	−32454.7	−1.83837
$679$	−13345.7	−0.754289
$680$	0	0
$681$	−43912.4	−2.47096
$682$	423.602	0.0237838
$683$	21781.9	1.22030	0.610148	−	0.792287i	$-0.291110\pi$
$683$	21781.9	1.22030	0.610148	+	0.792287i	$0.291110\pi$
$684$	10583.3	0.591611
$685$	0	0
$686$	3453.61	0.192215
$687$	−30009.2	−1.66655
$688$	−268.703	−0.0148899
$689$	−20902.5	−1.15577
$690$	0	0
$691$	−17943.9	−0.987871	−0.493935	−	0.869499i	$-0.664442\pi$
$691$	−17943.9	−0.987871	−0.493935	+	0.869499i	$0.664442\pi$
$692$	842.890	0.0463033
$693$	−1977.92	−0.108420
$694$	11480.5	0.627947
$695$	0	0
$696$	−695.210	−0.0378619
$697$	−3746.58	−0.203604
$698$	−18748.2	−1.01666
$699$	244.614	0.0132362
$700$	0	0
$701$	−32805.5	−1.76754	−0.883771	−	0.467920i	$-0.845004\pi$
$701$	−32805.5	−1.76754	−0.883771	+	0.467920i	$0.845004\pi$
$702$	28380.4	1.52586
$703$	−17705.3	−0.949883
$704$	101.198	0.00541769
$705$	0	0
$706$	7540.80	0.401986
$707$	13541.1	0.720320
$708$	16508.0	0.876286
$709$	37713.1	1.99767	0.998834	−	0.0482789i	$-0.0153736\pi$
$709$	37713.1	1.99767	0.998834	+	0.0482789i	$0.0153736\pi$
$710$	0	0
$711$	−68536.4	−3.61507
$712$	−634.858	−0.0334161
$713$	20644.5	1.08435
$714$	−7425.83	−0.389222
$715$	0	0
$716$	−8312.27	−0.433861
$717$	−52295.0	−2.72384
$718$	−7901.94	−0.410721
$719$	31029.2	1.60945	0.804725	−	0.593648i	$-0.202313\pi$
$719$	31029.2	1.60945	0.804725	+	0.593648i	$0.202313\pi$
$720$	0	0
$721$	40722.5	2.10345
$722$	−8202.07	−0.422783
$723$	62501.6	3.21502
$724$	16935.8	0.869357
$725$	0	0
$726$	−23372.7	−1.19482
$727$	−14941.8	−0.762256	−0.381128	−	0.924522i	$-0.624464\pi$
$727$	−14941.8	−0.762256	−0.381128	+	0.924522i	$0.624464\pi$
$728$	−13704.1	−0.697674
$729$	−26230.8	−1.33266
$730$	0	0
$731$	−285.497	−0.0144453
$732$	27329.6	1.37996
$733$	−14850.0	−0.748291	−0.374145	−	0.927370i	$-0.622064\pi$
$733$	−14850.0	−0.748291	−0.374145	+	0.927370i	$0.622064\pi$
$734$	−27165.2	−1.36606
$735$	0	0
$736$	4931.96	0.247003
$737$	117.843	0.00588983
$738$	−22206.6	−1.10764
$739$	18068.2	0.899391	0.449695	−	0.893182i	$-0.351533\pi$
$739$	18068.2	0.899391	0.449695	+	0.893182i	$0.351533\pi$
$740$	0	0
$741$	31872.9	1.58014
$742$	15044.1	0.744323
$743$	−24988.6	−1.23384	−0.616919	−	0.787027i	$-0.711619\pi$
$743$	−24988.6	−1.23384	−0.616919	+	0.787027i	$0.711619\pi$
$744$	9426.31	0.464497
$745$	0	0
$746$	−5690.52	−0.279283
$747$	−31495.6	−1.54265
$748$	107.523	0.00525593
$749$	−50249.0	−2.45135
$750$	0	0
$751$	−37959.1	−1.84440	−0.922202	−	0.386708i	$-0.873612\pi$
$751$	−37959.1	−1.84440	−0.922202	+	0.386708i	$0.873612\pi$
$752$	3597.06	0.174430
$753$	−14649.0	−0.708952
$754$	−1363.17	−0.0658404
$755$	0	0
$756$	−20426.2	−0.982663
$757$	7937.90	0.381120	0.190560	−	0.981676i	$-0.438970\pi$
$757$	7937.90	0.381120	0.190560	+	0.981676i	$0.438970\pi$
$758$	1142.13	0.0547281
$759$	2143.77	0.102522
$760$	0	0
$761$	28231.7	1.34481	0.672404	−	0.740184i	$-0.265262\pi$
$761$	28231.7	1.34481	0.672404	+	0.740184i	$0.265262\pi$
$762$	−19255.1	−0.915404
$763$	9647.87	0.457767
$764$	−11297.0	−0.534960
$765$	0	0
$766$	18018.2	0.849902
$767$	32369.0	1.52383
$768$	2251.94	0.105807
$769$	12843.5	0.602275	0.301137	−	0.953581i	$-0.402634\pi$
$769$	12843.5	0.602275	0.301137	+	0.953581i	$0.402634\pi$
$770$	0	0
$771$	20508.4	0.957967
$772$	12021.3	0.560435
$773$	30603.0	1.42395	0.711976	−	0.702204i	$-0.247800\pi$
$773$	30603.0	1.42395	0.711976	+	0.702204i	$0.247800\pi$
$774$	−1692.19	−0.0785846
$775$	0	0
$776$	4300.15	0.198926
$777$	73633.3	3.39972
$778$	2279.13	0.105027
$779$	−11573.9	−0.532321
$780$	0	0
$781$	660.978	0.0302838
$782$	5240.20	0.239628
$783$	−2031.83	−0.0927352
$784$	4375.21	0.199308
$785$	0	0
$786$	−35983.3	−1.63293
$787$	4230.46	0.191613	0.0958067	−	0.995400i	$-0.469457\pi$
$787$	4230.46	0.191613	0.0958067	+	0.995400i	$0.469457\pi$
$788$	2026.02	0.0915915
$789$	52495.9	2.36870
$790$	0	0
$791$	45801.5	2.05881
$792$	637.307	0.0285931
$793$	53587.9	2.39970
$794$	−3644.16	−0.162880
$795$	0	0
$796$	−13633.0	−0.607048
$797$	−3029.04	−0.134623	−0.0673113	−	0.997732i	$-0.521442\pi$
$797$	−3029.04	−0.134623	−0.0673113	+	0.997732i	$0.521442\pi$
$798$	−22939.8	−1.01762
$799$	3821.87	0.169222
$800$	0	0
$801$	−3998.09	−0.176361
$802$	27277.2	1.20099
$803$	1607.70	0.0706533
$804$	2622.33	0.115028
$805$	0	0
$806$	18483.1	0.807743
$807$	42730.3	1.86391
$808$	−4363.10	−0.189967
$809$	−17646.9	−0.766912	−0.383456	−	0.923559i	$-0.625266\pi$
$809$	−17646.9	−0.766912	−0.383456	+	0.923559i	$0.625266\pi$
$810$	0	0
$811$	24493.7	1.06053	0.530265	−	0.847832i	$-0.322092\pi$
$811$	24493.7	1.06053	0.530265	+	0.847832i	$0.322092\pi$
$812$	981.111	0.0424018
$813$	−9058.58	−0.390773
$814$	−1066.18	−0.0459087
$815$	0	0
$816$	2392.69	0.102648
$817$	−881.957	−0.0377672
$818$	−6797.60	−0.290553
$819$	−86303.0	−3.68214
$820$	0	0
$821$	−26560.5	−1.12907	−0.564535	−	0.825409i	$-0.690945\pi$
$821$	−26560.5	−1.12907	−0.564535	+	0.825409i	$0.690945\pi$
$822$	−15855.5	−0.672777
$823$	−16626.8	−0.704222	−0.352111	−	0.935958i	$-0.614536\pi$
$823$	−16626.8	−0.704222	−0.352111	+	0.935958i	$0.614536\pi$
$824$	−13121.3	−0.554734
$825$	0	0
$826$	−23296.9	−0.981359
$827$	7744.68	0.325645	0.162823	−	0.986655i	$-0.447940\pi$
$827$	7744.68	0.325645	0.162823	+	0.986655i	$0.447940\pi$
$828$	31059.6	1.30362
$829$	−7134.43	−0.298901	−0.149451	−	0.988769i	$-0.547750\pi$
$829$	−7134.43	−0.298901	−0.149451	+	0.988769i	$0.547750\pi$
$830$	0	0
$831$	−12418.3	−0.518393
$832$	4415.61	0.183995
$833$	4648.66	0.193357
$834$	37250.1	1.54660
$835$	0	0
$836$	332.160	0.0137416
$837$	27549.5	1.13769
$838$	−25247.5	−1.04076
$839$	−31789.3	−1.30809	−0.654046	−	0.756455i	$-0.726930\pi$
$839$	−31789.3	−1.30809	−0.654046	+	0.756455i	$0.726930\pi$
$840$	0	0
$841$	−24291.4	−0.995998
$842$	−1354.82	−0.0554514
$843$	−36983.2	−1.51100
$844$	−12097.1	−0.493364
$845$	0	0
$846$	22652.9	0.920593
$847$	32984.5	1.33809
$848$	−4847.39	−0.196297
$849$	36760.2	1.48599
$850$	0	0
$851$	−51961.0	−2.09307
$852$	14708.6	0.591442
$853$	17390.3	0.698044	0.349022	−	0.937115i	$-0.386514\pi$
$853$	17390.3	0.698044	0.349022	+	0.937115i	$0.386514\pi$
$854$	−38568.7	−1.54543
$855$	0	0
$856$	16190.8	0.646484
$857$	45015.1	1.79426	0.897132	−	0.441762i	$-0.145646\pi$
$857$	45015.1	1.79426	0.897132	+	0.441762i	$0.145646\pi$
$858$	1919.33	0.0763694
$859$	−12119.6	−0.481393	−0.240697	−	0.970600i	$-0.577376\pi$
$859$	−12119.6	−0.481393	−0.240697	+	0.970600i	$0.577376\pi$
$860$	0	0
$861$	48134.0	1.90523
$862$	−6973.94	−0.275561
$863$	10385.6	0.409652	0.204826	−	0.978798i	$-0.434337\pi$
$863$	10385.6	0.409652	0.204826	+	0.978798i	$0.434337\pi$
$864$	6581.55	0.259154
$865$	0	0
$866$	6590.42	0.258604
$867$	2542.23	0.0995832
$868$	−13302.8	−0.520193
$869$	−2151.04	−0.0839689
$870$	0	0
$871$	5141.88	0.200030
$872$	−3108.65	−0.120725
$873$	27080.7	1.04988
$874$	16188.0	0.626508
$875$	0	0
$876$	35775.8	1.37986
$877$	25002.9	0.962700	0.481350	−	0.876528i	$-0.340147\pi$
$877$	25002.9	0.962700	0.481350	+	0.876528i	$0.340147\pi$
$878$	30273.3	1.16364
$879$	9805.87	0.376273
$880$	0	0
$881$	25741.1	0.984381	0.492190	−	0.870488i	$-0.336196\pi$
$881$	25741.1	0.984381	0.492190	+	0.870488i	$0.336196\pi$
$882$	27553.4	1.05189
$883$	−25937.4	−0.988522	−0.494261	−	0.869314i	$-0.664561\pi$
$883$	−25937.4	−0.988522	−0.494261	+	0.869314i	$0.664561\pi$
$884$	4691.59	0.178501
$885$	0	0
$886$	114.542	0.00434326
$887$	21892.0	0.828707	0.414353	−	0.910116i	$-0.364008\pi$
$887$	21892.0	0.828707	0.414353	+	0.910116i	$0.364008\pi$
$888$	−23725.5	−0.896594
$889$	27173.6	1.02517
$890$	0	0
$891$	709.892	0.0266917
$892$	1725.30	0.0647617
$893$	11806.5	0.442430
$894$	−29334.7	−1.09743
$895$	0	0
$896$	−3178.04	−0.118494
$897$	93539.9	3.48184
$898$	8044.71	0.298948
$899$	−1323.26	−0.0490913
$900$	0	0
$901$	−5150.36	−0.190436
$902$	−696.961	−0.0257276
$903$	3667.91	0.135172
$904$	−14757.8	−0.542961
$905$	0	0
$906$	−52786.8	−1.93568
$907$	−40733.6	−1.49122	−0.745611	−	0.666382i	$-0.767842\pi$
$907$	−40733.6	−1.49122	−0.745611	+	0.666382i	$0.767842\pi$
$908$	−19967.8	−0.729796
$909$	−27477.1	−1.00260
$910$	0	0
$911$	24661.0	0.896876	0.448438	−	0.893814i	$-0.351981\pi$
$911$	24661.0	0.896876	0.448438	+	0.893814i	$0.351981\pi$
$912$	7391.48	0.268373
$913$	−988.499	−0.0358319
$914$	−6621.04	−0.239611
$915$	0	0
$916$	−13645.8	−0.492215
$917$	50781.2	1.82873
$918$	6992.90	0.251416
$919$	36860.4	1.32308	0.661541	−	0.749909i	$-0.269903\pi$
$919$	36860.4	1.32308	0.661541	+	0.749909i	$0.269903\pi$
$920$	0	0
$921$	70366.1	2.51753
$922$	−10554.4	−0.376995
$923$	28840.7	1.02850
$924$	−1381.40	−0.0491825
$925$	0	0
$926$	−25145.2	−0.892358
$927$	−82632.6	−2.92773
$928$	−316.125	−0.0111824
$929$	36545.1	1.29064	0.645320	−	0.763912i	$-0.276724\pi$
$929$	36545.1	1.29064	0.645320	+	0.763912i	$0.276724\pi$
$930$	0	0
$931$	14360.6	0.505532
$932$	111.230	0.00390931
$933$	−88816.8	−3.11654
$934$	6283.53	0.220132
$935$	0	0
$936$	27807.8	0.971076
$937$	21635.5	0.754324	0.377162	−	0.926147i	$-0.376900\pi$
$937$	21635.5	0.754324	0.377162	+	0.926147i	$0.376900\pi$
$938$	−3700.75	−0.128821
$939$	48845.8	1.69757
$940$	0	0
$941$	45144.9	1.56395	0.781977	−	0.623307i	$-0.214211\pi$
$941$	45144.9	1.56395	0.781977	+	0.623307i	$0.214211\pi$
$942$	−48324.7	−1.67145
$943$	−33966.8	−1.17297
$944$	7506.52	0.258810
$945$	0	0
$946$	−53.1100	−0.00182532
$947$	−11904.0	−0.408478	−0.204239	−	0.978921i	$-0.565472\pi$
$947$	−11904.0	−0.408478	−0.204239	+	0.978921i	$0.565472\pi$
$948$	−47866.6	−1.63991
$949$	70149.4	2.39952
$950$	0	0
$951$	−39891.0	−1.36021
$952$	−3376.67	−0.114956
$953$	290.324	0.00986832	0.00493416	−	0.999988i	$-0.498429\pi$
$953$	290.324	0.00986832	0.00493416	+	0.999988i	$0.498429\pi$
$954$	−30527.0	−1.03600
$955$	0	0
$956$	−23779.5	−0.804483
$957$	−137.410	−0.00464142
$958$	−20046.4	−0.676066
$959$	22375.9	0.753448
$960$	0	0
$961$	−11849.0	−0.397739
$962$	−46521.0	−1.55915
$963$	101963.	3.41197
$964$	28420.7	0.949551
$965$	0	0
$966$	−67323.3	−2.24233
$967$	−30479.4	−1.01360	−0.506801	−	0.862063i	$-0.669172\pi$
$967$	−30479.4	−1.01360	−0.506801	+	0.862063i	$0.669172\pi$
$968$	−10628.0	−0.352889
$969$	7853.45	0.260360
$970$	0	0
$971$	9696.60	0.320473	0.160236	−	0.987079i	$-0.448774\pi$
$971$	9696.60	0.320473	0.160236	+	0.987079i	$0.448774\pi$
$972$	−6415.67	−0.211711
$973$	−52568.9	−1.73205
$974$	8473.45	0.278754
$975$	0	0
$976$	12427.3	0.407569
$977$	47627.8	1.55962	0.779810	−	0.626016i	$-0.215315\pi$
$977$	47627.8	1.55962	0.779810	+	0.626016i	$0.215315\pi$
$978$	12608.5	0.412245
$979$	−125.481	−0.00409643
$980$	0	0
$981$	−19577.1	−0.637154
$982$	−16550.6	−0.537833
$983$	−40710.5	−1.32092	−0.660460	−	0.750862i	$-0.729639\pi$
$983$	−40710.5	−1.32092	−0.660460	+	0.750862i	$0.729639\pi$
$984$	−15509.3	−0.502458
$985$	0	0
$986$	−335.883	−0.0108486
$987$	−49101.3	−1.58350
$988$	14493.2	0.466691
$989$	−2588.35	−0.0832200
$990$	0	0
$991$	−4547.97	−0.145783	−0.0728916	−	0.997340i	$-0.523223\pi$
$991$	−4547.97	−0.145783	−0.0728916	+	0.997340i	$0.523223\pi$
$992$	4286.32	0.137188
$993$	29604.2	0.946085
$994$	−20757.4	−0.662360
$995$	0	0
$996$	−21996.8	−0.699795
$997$	−44017.5	−1.39824	−0.699121	−	0.715003i	$-0.746425\pi$
$997$	−44017.5	−1.39824	−0.699121	+	0.715003i	$0.746425\pi$
$998$	−21307.0	−0.675812
$999$	−69340.5	−2.19603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.4.a.u.1.5	yes	5
5.2	odd	4		850.4.c.o.749.6		10
5.3	odd	4		850.4.c.o.749.5		10
5.4	even	2		850.4.a.p.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
850.4.a.p.1.1	✓	5	5.4	even	2
850.4.a.u.1.5	yes	5	1.1	even	1	trivial
850.4.c.o.749.5		10	5.3	odd	4
850.4.c.o.749.6		10	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	850.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +8.79664 q^{3} +4.00000 q^{4} +17.5933 q^{6} -24.8284 q^{7} +8.00000 q^{8} +50.3809 q^{9} +1.58122 q^{11} +35.1866 q^{12} +68.9939 q^{13} -49.6568 q^{14} +16.0000 q^{16} +17.0000 q^{17} +100.762 q^{18} +52.5163 q^{19} -218.407 q^{21} +3.16244 q^{22} +154.124 q^{23} +70.3731 q^{24} +137.988 q^{26} +205.674 q^{27} -99.3137 q^{28} -9.87891 q^{29} +133.948 q^{31} +32.0000 q^{32} +13.9094 q^{33} +34.0000 q^{34} +201.524 q^{36} -337.139 q^{37} +105.033 q^{38} +606.915 q^{39} -220.387 q^{41} -436.813 q^{42} -16.7940 q^{43} +6.32489 q^{44} +308.247 q^{46} +224.816 q^{47} +140.746 q^{48} +273.450 q^{49} +149.543 q^{51} +275.976 q^{52} -302.962 q^{53} +411.347 q^{54} -198.627 q^{56} +461.967 q^{57} -19.7578 q^{58} +469.158 q^{59} +776.705 q^{61} +267.895 q^{62} -1250.88 q^{63} +64.0000 q^{64} +27.8189 q^{66} +74.5265 q^{67} +68.0000 q^{68} +1355.77 q^{69} +418.017 q^{71} +403.047 q^{72} +1016.75 q^{73} -674.277 q^{74} +210.065 q^{76} -39.2592 q^{77} +1213.83 q^{78} -1360.36 q^{79} +448.952 q^{81} -440.774 q^{82} -625.149 q^{83} -873.627 q^{84} -33.5879 q^{86} -86.9012 q^{87} +12.6498 q^{88} -79.3572 q^{89} -1713.01 q^{91} +616.494 q^{92} +1178.29 q^{93} +449.632 q^{94} +281.493 q^{96} +537.519 q^{97} +546.901 q^{98} +79.6634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 10 q^{2} + 5 q^{3} + 20 q^{4} + 10 q^{6} - q^{7} + 40 q^{8} + 54 q^{9} + 38 q^{11} + 20 q^{12} + 93 q^{13} - 2 q^{14} + 80 q^{16} + 85 q^{17} + 108 q^{18} - 50 q^{19} + 55 q^{21} + 76 q^{22} + 98 q^{23}+ \cdots + 1006 q^{99}+O(q^{100}) \) 5 * q + 10 * q^2 + 5 * q^3 + 20 * q^4 + 10 * q^6 - q^7 + 40 * q^8 + 54 * q^9 + 38 * q^11 + 20 * q^12 + 93 * q^13 - 2 * q^14 + 80 * q^16 + 85 * q^17 + 108 * q^18 - 50 * q^19 + 55 * q^21 + 76 * q^22 + 98 * q^23 + 40 * q^24 + 186 * q^26 + 365 * q^27 - 4 * q^28 + 102 * q^29 + 149 * q^31 + 160 * q^32 + 28 * q^33 + 170 * q^34 + 216 * q^36 + 414 * q^37 - 100 * q^38 + 289 * q^39 + 68 * q^41 + 110 * q^42 - 180 * q^43 + 152 * q^44 + 196 * q^46 + 958 * q^47 + 80 * q^48 + 734 * q^49 + 85 * q^51 + 372 * q^52 - 171 * q^53 + 730 * q^54 - 8 * q^56 + 1564 * q^57 + 204 * q^58 + 404 * q^59 + 650 * q^61 + 298 * q^62 - 694 * q^63 + 320 * q^64 + 56 * q^66 + 1792 * q^67 + 340 * q^68 + 1568 * q^69 + 283 * q^71 + 432 * q^72 + 294 * q^73 + 828 * q^74 - 200 * q^76 + 2284 * q^77 + 578 * q^78 + 951 * q^79 - 1355 * q^81 + 136 * q^82 - 1468 * q^83 + 220 * q^84 - 360 * q^86 + 2544 * q^87 + 304 * q^88 + 1666 * q^89 - 915 * q^91 + 392 * q^92 - 147 * q^93 + 1916 * q^94 + 160 * q^96 + 2304 * q^97 + 1468 * q^98 + 1006 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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